Advertisements
Advertisements
प्रश्न
Integrate the function in x log 2x.
Advertisements
उत्तर
Let `I = int x log 2x dx`
`= (log 2x) * x^2/2 - int d/dx (log 2x) (x^2)/2 dx`
`= log (2x)* x^2/2 - int 2/(2x) (x^2/2) dx + C`
`= x^2/2 log (2x) - 1/2 int x dx + C`
`= x^2/2 log (2x) - 1/2 * x^2/2 + C`
`= x^2/2 log (2x) - x^2/4 + C`
APPEARS IN
संबंधित प्रश्न
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
Integrate the function in (x2 + 1) log x.
Integrate the function in ex (sinx + cosx).
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Prove that:
`int sqrt(x^2 + a^2)dx = x/2 sqrt(x^2 + a^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c`
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Evaluate the following.
`int "x"^3 "e"^("x"^2)`dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
Evaluate: `int "dx"/(5 - 16"x"^2)`
Evaluate:
∫ (log x)2 dx
`int (sinx)/(1 + sin x) "d"x`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
Solve: `int sqrt(4x^2 + 5)dx`
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
`int1/sqrt(x^2 - a^2) dx` = ______
`inte^(xloga).e^x dx` is ______
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate:
`int e^(ax)*cos(bx + c)dx`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate `int tan^-1x dx`
If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv dx - int(d/dx u)(intv dx)dx`. Hence evaluate: `intx cos x dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate the following.
`int x^3 e^(x^2) dx`
